Initial normal covers in bi-Heyting toposes

Francis Borceux; Dominique Bourn; Peter Johnstone

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 4, page 335-356
  • ISSN: 0044-8753

Abstract

top
The dual of the category of pointed objects of a topos is semi-abelian, thus is provided with a notion of semi-direct product and a corresponding notion of action. In this paper, we study various conditions for representability of these actions. First, we show this to be equivalent to the existence of initial normal covers in the category of pointed objects of the topos. For Grothendieck toposes, actions are representable provided the topos admits an essential Boolean covering. This contains the case of Boolean toposes and toposes of presheaves. In the localic case, the representability of actions forces the topos to be bi-Heyting: the lattices of subobjects are both Heyting algebras and the dual of Heyting algebras.

How to cite

top

Borceux, Francis, Bourn, Dominique, and Johnstone, Peter. "Initial normal covers in bi-Heyting toposes." Archivum Mathematicum 042.4 (2006): 335-356. <http://eudml.org/doc/249831>.

@article{Borceux2006,
abstract = {The dual of the category of pointed objects of a topos is semi-abelian, thus is provided with a notion of semi-direct product and a corresponding notion of action. In this paper, we study various conditions for representability of these actions. First, we show this to be equivalent to the existence of initial normal covers in the category of pointed objects of the topos. For Grothendieck toposes, actions are representable provided the topos admits an essential Boolean covering. This contains the case of Boolean toposes and toposes of presheaves. In the localic case, the representability of actions forces the topos to be bi-Heyting: the lattices of subobjects are both Heyting algebras and the dual of Heyting algebras.},
author = {Borceux, Francis, Bourn, Dominique, Johnstone, Peter},
journal = {Archivum Mathematicum},
keywords = {semi-abelian category},
language = {eng},
number = {4},
pages = {335-356},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Initial normal covers in bi-Heyting toposes},
url = {http://eudml.org/doc/249831},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Borceux, Francis
AU - Bourn, Dominique
AU - Johnstone, Peter
TI - Initial normal covers in bi-Heyting toposes
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 4
SP - 335
EP - 356
AB - The dual of the category of pointed objects of a topos is semi-abelian, thus is provided with a notion of semi-direct product and a corresponding notion of action. In this paper, we study various conditions for representability of these actions. First, we show this to be equivalent to the existence of initial normal covers in the category of pointed objects of the topos. For Grothendieck toposes, actions are representable provided the topos admits an essential Boolean covering. This contains the case of Boolean toposes and toposes of presheaves. In the localic case, the representability of actions forces the topos to be bi-Heyting: the lattices of subobjects are both Heyting algebras and the dual of Heyting algebras.
LA - eng
KW - semi-abelian category
UR - http://eudml.org/doc/249831
ER -

References

top
  1. Borceux F., When is Ω a cogenerator in a topos?, Cahiers Topol. Géom. Diff. 16 (1975), 3–15. (1975) Zbl0311.18006MR0382393
  2. Borceux F., Handbook of Categorical Algebra 3: Categories of Sheaves, Encyclopaedia Math. Appl. 52 (1994). (1994) Zbl0911.18001MR1315049
  3. Borceux F., A survey of semi-abelian categories, In: Galois theory, Hopf Algebras, and Semi-abelian Categories, Fields Inst. Commun. 43 (2004), 27–60. Zbl1067.18010MR2075580
  4. Borceux F., Bourn D., Mal’cev, Protomodular, Homological and Semi-abelian Categories, Math. Appl. 566 (2004). Zbl1061.18001MR2044291
  5. Borceux F., Bourn D., Split extension classifier and centrality, to appear in the Proceedings of the Streetfest 2005. Zbl1133.18002MR2342823
  6. Borceux F., Janelidze G., Kelly G. M., Internal object actions, Comment. Math. Univ. Carolin. 46 (2005), 235–255. Zbl1121.18004MR2176890
  7. Borceux F., Janelidze G., Kelly G. M., On the representability of actions in a semi-abelian category, Theory Appl. Categ. 14 (2005), 244–286. Zbl1103.18006MR2182676
  8. Bourn D., Normal functors and strong protomodularity, Theory Appl. Categ. 7 (2000), 206–218. Zbl0947.18004MR1766393
  9. Bourn D., A categorical genealogy for the congruence distributive property, Theory Appl. Categ. 8 (2001), 391–407. Zbl0978.18005MR1847038
  10. Bourn D., Protomodular aspects of the dual of a topos, Adv. Math. 187 (2004), 240–255. MR2074178
  11. Bourn D., Janelidze G., Protomodularity, descent and semi-direct products, Theory Appl. Categ. 4 (1998), 37–46. (1998) MR1615341
  12. Janelidze G., Márki L., Tholen W., Semi-abelian categories, J. Pure Appl. Alg. 168 (2002), 367–386. Zbl0993.18008MR1887164
  13. Johnstone P. T., Stone Spaces, Cambridge Stud. Adv. Math. No. 3 (1982). (1982) Zbl0499.54001MR0698074
  14. Johnstone P. T., Sketches of an Elephant: a Topos Theory Compendium, volumes 1–2, Oxford Logic Guides 43–44 (2002). Zbl1071.18002MR1953060
  15. Mac Lane S., Categories for the Working Mathematician, Graduate Texts in Math. No. 5 (1971; revised edition 1998). (1971) Zbl0232.18001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.